1. theorems about
LINEAR MAPPINGS

2. T is a linear mapping from a vector space V into a vector space WU x x T y y T
the range
the domain

3. T is a linear mapping from a vector space V into a vector space WU x x T y y T x T y + x y +
the range x y + T( ) = T
the domain

4. THEOREM 1
If T is a linear mapping from V into W then
T( ) = V W

5. T is a linear mapping from a vector space V into a vector space Wx T U V W
the range
T( ) = V W
the domain

6. T( ) T( ) T( ) + x = x + x = linear mappingV
T( ) + V x =
linear mapping
must be the identity in the range

7. THEOREM 2
If T is a linear mapping from V into W then
T is one to one
if and only if
Null Space of T = { 0 }

8. T is a linear mapping from a vector space V into a vector space Wx T
null space U V W
If T is one to one
the range
the null space contains
only zero.
the domain

9. T is a linear mapping from a vector space V into a vector space W
If the null space contains
only zero. x T
null space U
and = x T y
then = 0 x T y V W
linear x y - x
then T( ) = 0 y x
the range y x y -
then
is in the null space x y -
then
= 0
the domain x y =
then

10. T is a linear mapping from a vector space V into a vector space W
If the null space contains
only zero. x T
null space U
and = x T y
then = 0 x T y V W
linear x y - x
then T( ) = 0 y x y
the range x y -
then
is in the null space x y -
then
= 0
the domain x y =
then

11. T is a linear mapping from a vector space V into a vector space W
If the null space contains
only zero. x T
null space U
and = x T y
then = 0 x T y V W
linear x y - x
then T( ) = 0 y x y y
the range x y -
then
is in the null space x y -
then
= 0
the domain x y =
then

12. THEOREM 3
If T is a linear mapping from V into W then
The Null Space of T is a vector space

13. T is a linear mapping from a vector space V into a vector space W
Let a and b be any two vectors in the null space V W a
T( ) = 0 b
T( ) = 0 x T
null space U a
T( ) + b
T( ) = 0 a b W a T( b
) = 0
a +
T is linear
the range b
is in the nullspace a
the domain

14. THEOREM 4
If T is a linear mapping from V into W then
The Range of T is a vector space

15. T is a linear mapping from a vector space V into a vector space Wx T y
and
are in the range of T V W x T x y
and
are in the domain
null space U x x T
x + y y T x y V W y
x +
y ) T(
x + y
is in the domain
the range
x +
y ) is in the range T(
T is linear
the domain x T y +
is in the range of T

16. THEOREM 5
If T is a linear mapping from V into W then
the dimension of the DOMAIN =
the dimension of the RANGE + the dimension of the NULLSPACE

17. T is a linear mapping from a vector space V into a vector space W
is a basis for the null space V W
is a basis for the domain x T
null space U x W y
the range
the domain
is a basis for the range